MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvopab4ndm Structured version   Visualization version   GIF version

Theorem fvopab4ndm 7021
Description: Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)
Hypothesis
Ref Expression
fvopab4ndm.1 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
Assertion
Ref Expression
fvopab4ndm 𝐵𝐴 → (𝐹𝐵) = ∅)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem fvopab4ndm
StepHypRef Expression
1 fvopab4ndm.1 . . . . 5 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
21dmeqi 5898 . . . 4 dom 𝐹 = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
3 dmopabss 5912 . . . 4 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
42, 3eqsstri 4011 . . 3 dom 𝐹𝐴
54sseli 3973 . 2 (𝐵 ∈ dom 𝐹𝐵𝐴)
6 ndmfv 6920 . 2 𝐵 ∈ dom 𝐹 → (𝐹𝐵) = ∅)
75, 6nsyl5 159 1 𝐵𝐴 → (𝐹𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1533  wcel 2098  c0 4317  {copab 5203  dom cdm 5669  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-dm 5679  df-iota 6489  df-fv 6545
This theorem is referenced by:  fvmptndm  7022
  Copyright terms: Public domain W3C validator